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After studying some foundation of Gromow-Witten invariants, I now would like to see an explicit computation. I heard that one should first take a look at the total space of $\mathcal{O}(-1)^{\oplus2}$ over $\mathbb{P}^1$ or the total space of the canonical bundle of Fano surface (local Calabi-Yau). They can be worked out very explicitly via equivariant cohomology and localization. (Or there may be more tractable examples)

Could someone kindly suggest a paper or lecture note where I can start learning these examples and technique? Any suggestion is welcome.

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Gromov-Witten classes, quantum cohomology, and enumerative geometry (by Kontsevich & Manin)

J-holomorphic curves and symplectic topology (by McDuff & Salamon)

A tutorial on quantum cohomology (by Givental)

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Thank you for the suggestion, Chris. The Givental's paper looks very good with Fano examples. I am now more interested in CY examples (I should have emphasize this point in my question). I don't have the first two books at hand now and cannot say anything, but if my memory serves they don't have much concrete examples but theories. Anyways, I appreciate your answer. Thanks! – Daniel Sep 12 '12 at 19:13
No they have examples. (and you can quickly download them on the internet) – Chris Gerig Sep 12 '12 at 23:00

Hope this is among the lines of what you're looking for.

The exercises are a very valuable part of this book, for it contains a handful of nice exercises.

As for lecture notes, Renzo Cavalieri has some nice notes you might find helpful. He knows a good deal localization.

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